Partial linear space

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A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space. The notion is equivalent to that of a linear hypergraph.

## Definition

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Let  S = ( P , L , I ) {\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})} an incidence structure, for which the elements of  P {\displaystyle {\mathcal {P}}} are called points and the elements of  L {\displaystyle {\mathcal {L}}} are called lines. S is a partial linear space, if the following axioms hold:

* any line is incident with at least two points
* any pair of distinct points is incident with at most one line

If there is a unique line incident with every pair of distinct points, then we get a linear space.

## Properties

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The De Bruijn–Erdős theorem shows that in any finite linear space  S = ( P , L , I ) {\displaystyle S=({\mathcal {P}},{\mathcal {L}},{\textbf {I}})} which is not a single point or a single line, we have  | P | ≤ | L | {\displaystyle |{\mathcal {P}}|\leq |{\mathcal {L}}|} .

## Examples

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* Projective space
* Affine space
* Polar space
* Generalized quadrangle
* Generalized polygon
* Near polygon

## References

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* Shult, Ernest E. (2011), Points and Lines, Universitext, Springer, doi:10.1007/978-3-642-15627-4, ISBN 978-3-642-15626-7.

* Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press 1986, ISBN 0-521-31857-2, p. 1-22
* Lynn Batten and Albrecht Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.
* Eric Moorhouse: Incidence Geometry. Lecture notes (archived)

## External links

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* partial linear space at the University of Kiel
* partial linear space at PlanetMath